Uniformly convergence on a set implies a smaller set

Question

Here is a product, that looks like this $\prod_{n=1}^{\infty}p_n(z)$. There are two questions:

• If you have shown that the product is uniformly convergent on a compact subset of $\mathbb{C}$, is this product then defined on $\mathbb{C}$? Or does it require to show that it is uniformly on every compact subsets?
• If you have shown that the infinite product is uniformly convergent on every compact subset of $\mathbb{C}$ does it imply that it is uniformly convergent on every compact subset of $\mathbb{C}\setminus B$, where $B$ is any set?

Answer 1

As for the first question, you'll have to show it's convergent on every compact subset of $\mathbb{C}$. The identity theorem does not extend to uniform convergence...

Since every compact subset of $\mathbb{C}\backslash B$ can be obtained by removing $B$ from a compact subset of $\mathbb{C}$ the answer to the second question is yes.